In triangle PQR, ∠P = 55° and QR = 18 cm. In which of the following cases,…

In any triangle, the three interior angles sum to 180°, and by the Law of Sines each side is proportional to the sine of the angle opposite it: side / sin(opposite angle) is the same for all three sides. A triangle is obtuse when one interior angle exceeds 90°, and scalene when all three sides — equivalently, all three angles — are mutually unequal. Equal sides always correspond to equal opposite angles, so setting one side equal to another whose opposite angle is already fixed forces their opposite angles to match too — and if the given angles do not match, no such triangle can exist at all.

1. ∠P is fixed at 55°, so for any assumed ∠R, ∠Q = 180° − 55° − ∠R = 125° − ∠R.

2. QR = 18 cm is the side opposite ∠P (= 55°). If another named side is also set equal to QR (= 18 cm), that would force its opposite angle to equal ∠P; if the case's own stated angle does not match ∠P, the configuration is not a valid triangle at all.

3. For ∠R = 15°: ∠Q = 125° − 15° = 110°, which is greater than 90°, so the triangle is obtuse.

4. The side condition here is PR > 18 cm — a strict inequality, not an equality — so PR is never forced equal to QR, and the angle-matching contradiction that rules out the other cases does not apply; the triangle can remain scalene.

Applying the Law of Sines independently: PR = QR × sin(∠Q) / sin(∠P) = 18 × sin 110° / sin 55° ≈ 18 × 0.9397 / 0.8192 ≈ 20.65 cm, confirming PR > 18 cm as stated.

Similarly, PQ = QR × sin(∠R) / sin(∠P) = 18 × sin 15° / sin 55° ≈ 18 × 0.2588 / 0.8192 ≈ 5.69 cm. The three sides 18 cm, 20.65 cm and 5.69 cm are all different, confirming the triangle is scalene as well as obtuse.

So the case ∠R = 15° with PR > 18 cm is the only one that produces an obtuse scalene triangle.